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The problem presented here involves finding the gross output needed to meet the final demand in a basic input-output model. Given the transaction matrix and the final demand for agriculture and industry, we will calculate the total gross output for each sector.

The table provided shows that:

- \( x_1 \) = Total output for the Agriculture sector
- \( x_2 \) = Total output for the Industry sector

The transaction matrix is:

\[
\begin{bmatrix}
300 & 600 \\
400 & 1200 \\
\end{bmatrix}
\]

Where:
- 300 units of agriculture are used in the production of agriculture.
- 600 units of agriculture are used in the production of industry.
- 400 units of industry are used in the production of agriculture.
- 1200 units of industry are used in the production of industry.

The final demand vector is:
\[
\begin{bmatrix}
100 \\
400 \\
\end{bmatrix}
\]

The equation to calculate the total output of the sectors can be written as:

\[
x_1 = 300 + 600 + 100 = 1000 \quad \text{(Agriculture sector total output)}
\]
\[
x_2 = 400 + 1200 + 400 = 2000 \quad \text{(Industry sector total output)}
\]

Therefore, the total gross output required to meet the final demand of 200 units of agriculture and 800 units of industry is:

\[
x_1 = 1000, \quad x_2 = 2000
\]

Would you like further explanation or details?

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Here are five related questions:
1. How do input-output models contribute to understanding interdependence in sectors?
2. What are the limitations of this type of input-output analysis?
3. How would the total output change if the final demand increased?
4. How can this model be applied to a real economy's sectors?
5. What other mathematical methods are used in economics to measure sectoral output?

**Tip:** When working with input-output matrices, make sure to correctly account for all intermediate and final demands to get accurate sector outputs.

Given the following transaction matrix, find the gross output to meet the final demand of 200 units of Agriculture and 800 units of Industry. Producing Sector: Agriculture and Industry with respective values in a matrix form.

Learn how to find the gross output for Agriculture and Industry using an input-output matrix. This problem involves calculating the total output required to meet a specific final demand of 200 units for Agriculture and 800 units for Industry. Suitable for college-level students studying Linear Algebra and Economics.

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